UH MATH 2415 Calculus III: Syllabus, Topics and Practice

UH MATH 2415 (Calculus III) is the third course in the University of Houston calculus sequence and was previously numbered MATH 2433. It lists MATH 2414 as its prerequisite. This guide maps the department's published syllabus to lessons you can practice with instant feedback.

What UH MATH 2415 covers

The UH Department of Mathematics course guideline describes MATH 2415 as the calculus of functions of several variables: calculus of vector-valued functions, partial differentiation, multiple integrals. Its curriculum details add vectors and vector-valued functions, Lagrange multipliers, Jacobians, and line integrals with Green's Theorem, the Divergence Theorem and Stokes' Theorem. The learning outcomes also mention curvature, velocity, acceleration and torsion for vector-valued functions.

The syllabus is organized in six chapters:

1. Vectors: Cartesian space coordinates, vectors, the dot and cross product, lines and planes.
2. Vector calculus: vector functions, differentiation formulas, curves, arc length, curvilinear motion and curvature.
3. Functions of several variables: graphs, level curves and surfaces, limits and continuity, partial derivatives.
4. Gradients, extreme values, differentials: the gradient and directional derivatives, chain rules, tangent planes, local and absolute extremes, Lagrange multipliers, reconstructing a function from its gradient.
5. Double and triple integrals: repeated integrals, polar, cylindrical and spherical coordinates, Jacobians and change of variables.
6. Line and surface integrals: the Fundamental Theorem for line integrals, Green's Theorem, parameterized surfaces, surface integrals, the Divergence Theorem and Stokes's Theorem.

MATH 2415 topic map

MATH 2415 in the UH sequence

Example problems at the MATH 2415 level

1. A tangent plane. For z = x^2 + y^2 at (1, 1, 2):

2. A Jacobian. For polar coordinates x = r\cos\theta, y = r\sin\theta, the Jacobian is r, which is why dA = r\,dr\,d\theta.

3. Green's Theorem. For \mathbf{F} = \langle -y, x \rangle around the unit circle:

How to do well in MATH 2415

• Master vectors in the first chapter. Dot and cross products show up again in tangent planes, flux and Stokes' Theorem.
• Sketch level curves and regions before computing anything.
• Choose coordinates to match the shape in double and triple integrals.
• Make a one-page theorem sheet with what each of Green's, Stokes' and the Divergence Theorem says and when it applies, then test yourself with vector calculus practice.
• Use timed practice exams before each test.

Tip: before parameterizing a line integral, check if the field is a gradient. The syllabus covers reconstructing a function from its gradient for exactly this reason.